Application of humanitarian last mile distribution model Open Access

Historical data indicate that the total number of natural disasters has dramatically risen over the last ten years (Tang, 2006). According to Thomas and Kopczak (2005), they are even expected to increase another fivefold over the next 50 years, as ascribable to many different factors like global warming, population growth rate, urbanization, residential densification, economic and financial global contingencies, natural resources immoderate use and depletion, etc.

Due to these reasons, offering timely and necessary aid to those in need through efficient humanitarian supply chains is a major challenge and logistics acts as a strategic role, as expressed by Trunick (2005).

In fact, the goal of humanitarian operations has been defined by Tomasini and Van Wassenhove (2004) as “a successful humanitarian operation [that] mitigates the urgent needs of a population with a sustainable reduction of their vulnerability in the shortest amount of time and with the least amount of resources.” Indeed, one of the most critical tasks during the humanitarian operations, after a natural catastrophe, is to manage and execute all the logistics activities effectively and efficiently, especially in the last mile distribution problem.

Transportation is a key element of delivery and in many NGOs, UN agencies, and other humanitarian organizations, vehicle fleets represent the second largest overhead cost after staff expenses (Disparte, 2007). Hence, in this complex environment a growth in terms of efficiency has not just to be seen as cost saving but also as a service level increase and therefore a better performance in terms of social costs as suggested by the literature.

The particular “humanitarian environment” makes logistics a challenging field. Humanitarian logistics has the challenge of allocating scarce resources to complex operations in the most efficient way (Van Wassenhove and Pedraza Martinez, 2012), while considering invisible and/or qualitative factors (Van Wassenhove, 2006), under severe restrictions and random and imprecise information (Barbarosoglu and Arda, 2004).

As previously mentioned, notable attention has been paid to the distribution problems, where traditional approaches fail due to all the particularities of humanitarian operations.

Indeed, in the last mile distribution problems, not just economic factors, but also social factors are significant in the relief operations, where the purpose “is to rapidly provide the appropriate emergency supplies to people affected by natural and manmade disasters so as to minimize human suffering and death” (Balcik et al., 2008).

The main objective of the research concerns the development and application of a modified last mile distribution model to a real case study, starting from the collection of data and estimation of item demands, as well as real application of the model, and finally carrying out a sensitivity study of logistic costs and percentage of shortage varying the fleet, its capacity, and the introduction of co-transportation modality.

The applied model is a conceptual evolution of the one developed by Balcik et al. (2008), whose authors reported only a simple numerical example.

First, the improved model is proposed in order to study the logistic costs in function of the available vehicles, with no shortage permitted. The second part covers an important analyzed variable: the possibility of co-transportation, seen as “different conveyers/suppliers transport their goods in only one truck” (Hageback and Segerstedt, 2004). This kind of distribution has often been highlighted by the humanitarian experts in conferences and interviews as contributing to improved performance of operations at the same time. The last part deals with the situation of shortage in function of limited capacity of fleet, in order to understand how the service is related to the logistic systems.

Section 2 reviews the most relevant scientific contributions about routing problems in the humanitarian field. Section 3 regards the modified last mile distribution problem. In particular, initially the minimal theoretic number of vehicles has been estimated – which is required as the first data – and then two different models have been introduced: the modified last mile distribution model under no-shortage constraint and the model under limited resource capacity. Further, the Haitian case is presented with a focus on the data. In this section the application of the first model, the impact of co-transportation, and the impact of limited resource capacity, considered in the second model, are presented. Finally, the results and future researches are discussed.

In this section we introduce both the main humanitarian topics and the relevant literature in order to highlight how humanitarian operations supporting the planning and execution of the disaster relief constitute a relatively new field (Kovács and Spens, 2007).

According to Van Wassenhove (2006), in humanitarian operations there are many areas that can be successfully explored by academics and in operations research. All of these areas can be associated with the supply chain management topic and it is possible to find more sub topics, which are specific to improvement of the humanitarian supply chain performances. Some examples of these issues, as suggested by Van Wassenhove (2006), are the last mile problem, the partnership between humanitarian and private sector together with the project management, risk management, systems and technology design and optimization, process standardization, performance measurement and control.

At the same time, as widely suggested in the literature, the humanitarian operation can be considered as having different stages, each of which has its own objectives (Kovács and Spens, 2007; Charles, 2010; Peretti, 2011). If we consider the characterization by Holguìn-Veras et al. (2012), where the humanitarian operation is split into regular humanitarian logistics (R-HL) and post-disaster humanitarian logistics, the authors present the main differences between these two phases and the commercial logistics. According to this research, it has been decided to place the examination in the development phase of the operation, namely in the R-HL. This choice has been made because the post-disaster phase is considered too messy, and so it is not possible to have clear data about it. These data are fundamental for the application of the model below. In fact, in the literature, the high difficulty of acquiring reliable data has been underlined (Van Wassenhove and Pedraza Martinez, 2012) in the slogan “data is the beast” (Lee, 2010).

The present research examines the last mile problem and, in particular, the logistic facility management, coupled with material distribution modalities, is studied. In the scientific literature, the last mile problem is widely faced in order to improve performances because “inefficient deliveries in this last mile of supply chain have led numerous business collapses as well as a substantial increase in delivery costs” (Boyer et al., 2009). For this reason in 2009, Taubenböck et al. present an interdisciplinary approach for the last mile preparation for a potential disaster while Van Hentenryck et al. (2010), always in last mile, face the single commodity allocation problem for disaster recovery. This work focusses on the transportation because it is considered as a cornerstone of last mile distribution problem, as Balcik et al. present in 2008. Field vehicles are used to transport humanitarian staff, aid items, and materials in an environment where no vehicles means no aid (Van Wassenhove and Pedraza Martinez, 2012). In a complex environment, usually under conditions requiring the allocation of scarce resources (Altay and Green, 2006), according to Stapleton et al. (2009), “the potential saving from implementing operations research models can be used in improving the social welfare of populations need,” so minimizing human suffering and the loss of lives. At the same time, when considering the importance of transportation, which is seen as an important key to reduce the social, economic, and environmental impacts (Berkoune et al., 2012), it is possible to understand how the last mile problem faces both the economic and social problems associated with the situation after disasters.

There are some models associated with distribution in humanitarian operations. Knott (1987) considered the last mile delivery of food items from a distribution center to a number of camps. Haghani and Oh (1996) introduced a multi-commodity and multi-modal network flow in the discussion, while Barbarosoglu et al. (2002) decomposed the problem hierarchically into two sub-problems, focussing on a deterministic single-period model to optimize helicopter distribution. In 2004 Barbarosoglu and Arda adopted a stochastic multi-objective single-period model with an example of its application using data from a real earthquake in Turkey in 1999. Finally, in 2008 Balcik et al. developed a model that considered different relief items with different demand characterizations and criticality of supplies in resource allocation. Further, they considered the complexity of the environment and the uncertainty of the humanitarian operations. Other authors can be found in the literature review by Caunhye et al. (2012), who consider the optimization models about relief distribution.

The model applied in this script optimizes resource allocation and vehicle routing decisions in a real case, the well-known Haitian case, on a number of test problems.

Typically, last mile distribution problem concerns the study of distribution in the last part of the supply chain. In this particular situation, the most important goal is to improve performances in order to supply aid to the beneficiaries without inefficiencies, due to the high cost of unsatisfied demand (Balcik et al., 2008; Taubenbock et al., 2009).

Known the amount of demand, the available route networks and the different transportation modalities, the problem is to define the most convenient distribution plan, meant as relief supply allocation, vehicle delivery scheduling, and vehicle routing, as defined by Balcik et al. (2008). For more details, the readers could see the explanation included in Balcik et al. (2008).

The two models applied in this research are a conceptual evolution of the mixed integer model recently developed by Balcik et al. (2008). The aim of these models was to develop an efficient resource allocation mechanism that minimizes suffering, while achieving equity in relief aid distribution among affected areas.

In this present work, the model presented by Balcik et al. (2008) has been modified in order to maximize the beneficiaries and minimize transportation costs, by optimizing resource allocation and vehicle routing decisions.

The main differences of these two models from the one proposed by Balcik et al. (2008) are the following constraints:

1. (1) no shortage is accepted (present in the first model below); and

2. (2) limited resources capacity (present in the second model below).

Moreover, Balcik et al. (2008) report only a simple numerical example. In comparison, this paper aims to apply a mathematical model using real assessed data from the Haitian case study.

The purpose of the evolution of the model is to optimize material deliveries involved in a real relief operation, also considering co-transportation modalities with different costs, capacities, and routes.

The first model considers the possibility of delivering aid using two kinds of vehicles (trucks and helicopters) that have different costs and transportation times. The zero shortage condition is a model constraint, and the cost of the fleet is expressed in function of the kind of vehicle applied during the relief operation. Then, we explore the impact of variations in available logistics assets, modifying the objective function and relative constraints.

In the next stage, the second model includes a different objective function to minimize the shortage. Indeed, the aim is to maximize the level of service to people receiving aid from the organization, by varying the number and type of used vehicles.

Sets

T set of days t in the planning horizon; length of the planning horizon

K set of trucks k

R k set of routes r k for trucks k

H set of helicopters h

R h set of routes r h for helicopters h

E set of demand types e: E={1,2}

N set of all demand locations i

N _r k set of demand locations i visited on route r k∈R k

N _r h set of demand locations i visited on route r h∈R h

Routing parameters

C _r k k cost of route r k for truck k∈K

C _r h h cost of route r h for helicopter h∈H

q _k capacity of truck k∈K (amount of demand)

q _h capacity of helicopter h∈H (amount of demand)

T _r k k duration of route r k∈R k for truck k∈K

T _r h h duration of route r h∈R h for helicopter h∈H

M _k available time of truck k∈K

M _h available time of helicopter h∈H

Demand parameters

(Inline Equation 1) demand of type e at location i∈N on day t∈T (amount of demand per day)

(Inline Equation 2) priority index of e∈E relief supplies to deliver to location i∈N on day t∈T

(Inline Equation 3) amount of type e∈E relief supplies arriving to the truck local distribution center (LDC) at the beginning of day t∈T

(Inline Equation 4) amount of type e∈E relief supplies arriving to the helicopter LDC at the beginning of day t∈T

Routing decision variables

(Equation 5) 

(Equation 6) 

(Equation 7) 

(Equation 8) 

Delivery decision variables

(Inline Equation 9) amount of demand of type e∈E delivered to location i∈N _r on day t∈T by vehicle k∈K via route r k∈R k

(Inline Equation 10) amount of demand of type e∈E delivered to location i∈N _rh on day t∈T by helicopter h∈H via route r h∈R h

(Inline Equation 11) percentage of unsatisfied type e demand at location i∈N by day t∈T

As discussed in the previous sections, the number of available vehicles is one of the most important aspects to consider when seeking to satisfy all the demand points without incurring a shortage.

Since the accuracy and the availability of data about demand, LDC, and locations are very low, it is useful to estimate the minimal number of vehicles based on assessed data at the beginning of the study, in order to design the correct dimension of vehicles fleet.

We introduce a simple formulation to assess the minimal theoretical number of vehicles using the following simple estimated parameters:

• (Inline Equation 12) demand of each type of item;

• (Inline Equation 13) the average route in time for trucks; and

• (Inline Equation 14) the average route in time for helicopters.

Typically, the managers of humanitarian organizations estimate the first parameter by multiplying the expected beneficiaries of each analyzed location and per capita consumption of each item.

As regards route data, they are assessed studying the road and air network composed of the positions of beneficiaries’ locations and LDCs.

After assessing these parameters in the first phase of the last mile distribution problem, the minimal theoretical number of vehicles is computed as follows:

(Equation 15) (Equation 16) 

These formulas assist in making an estimation of the number of vehicles, both trucks (1) and helicopters (2), that are necessary to satisfy the entire demand.

The formulation of the first problem model is: (Equation 17) 

Constraints:

(Equation 18) (Equation 19) (Equation 20) (Equation 21) (Equation 22) (Equation 23) (Equation 24) (Equation 25) (Equation 26) (Equation 27) (Equation 28) (Equation 29) (Equation 30) 

The objective function (3) minimizes costs associated with trucks and helicopters and considers both a truck and helicopter delivery.

The assumptions of the problem are based on the model introduced by Balcik et al. (2008), who consider the typical routing problem constraints in the humanitarian field.

In more detail, the first constraint (4) means that the unsatisfied demand has to be zero. This constraint allows to be studied at this stage just the costs and not the shortage variation.

Constraint (5) finds the amount of unsatisfied demand and it implies that the demand of a location is completely satisfied by the delivered amount expressed by the next constraint (6) that indicated the number of items e delivered location i on day t.

Constraints (7) and (8) consider the capacity limit for trucks and helicopters, while (9) and (10) concern the available time in the time window for both of the vehicles, and M _k and M _h are the number of minutes available in the day for each kind of transportation mode. Constraints (11) and (12) are the number of items present in truck and helicopter LDCs. Constraints (13) and (14) are non-negativity constraints, while (15) and (16) define the binary routing variable.

The aim of this part of the research is to study the model behavior in the presence of limited resource capacity in humanitarian logistics operations, so to understand the variation of the shortage and the relative cost for all the e items considered in the whole planning horizon in function of the number of the available trucks and helicopters, calculated in the range of values assessed using formulas (1) and (2).

In this case the objective function changes as follows:

(Equation 31) 

it is subjected to almost all the constraints seen in the previous part, without considering the constraint number (4), but with one more constrain indicated as follows:

(Equation 32) 

The constrain (18) suggests that the level of shortage can vary from 0, in the case that every beneficiary is satisfied, and the demand of the item e at the node i at the time t.

According to the United Nations Secretary-General report, which was published on September 2, 2011, the earthquake that hit Haiti in January 2010, measuring 7.2 on the Richter scale, affected almost 3.5 million people. It has been estimated by the Government of Haiti that the earthquake killed 222,570 and injured another 300,572 people. At least 188,383 houses were badly damaged and 105,000 were destroyed by the earthquake. Almost 60 percent of the Government and administrative buildings, 80 percent of schools in Port-au-Prince, and 60 percent of schools in the South and West Departments were destroyed or damaged. The total related loss has been estimated at $7.8 billion, more than 120 percent of Haiti's 2009 gross domestic product.

For these reasons, in order to meet the basic needs of the affected population, after the earthquake hit Haiti, several relief teams arrived and started their work. United Nations agencies, for example, as well as the International Federation of the Red Cross and Red Crescent Society started to prepare the deployment of teams and humanitarian assistance.

The organizations that will be considered in this article are World Food Program (WFP) and UNICEF, while most of the data used come from WFP documents (WFP, 2010) or interviews with WFP's operators. The WFP is a United Nations agency that is focussed on food planning and food deliveries to nutritionally vulnerable groups. Indeed, it is the leader of the food cluster, the logistic cluster, and the Emergency Telecommunications cluster. UNICEF, instead, is the leader of the Education, Water Sanitation Hygiene, and Nutrition clusters (for more information about the cluster see Jahre and Jensen, 2010 and the United Nation web site). For both the United Nations organizations the overall objective is to “save lives and protect livelihoods in emergencies” in their own fields.

In a disaster such as that in Haiti there are many items that have to be delivered to the vulnerable groups, including tents, blankets, tarpaulins, jerry cans, mosquito nets, food, and hygiene kits, as well as more cumbersome ones like kitchens and other materials for reconstruction. Here, hygiene kits (called “item1”) and food (called “item2”) will be taken into consideration (they are homogeneous items for shape and size). This choice has been supported by the documents analysis and by the fact that they are provided by two different organizations: UNICEF supplies hygiene kits, while WFP delivers food consumption. Thus, it is possible to understand what should happen when they are provided at the same time with a good level of coordination between organizations.

As argued by Balcik et al. (2008), during the relief operations there are some features that can bring more problems than during commercial distribution. These characteristics are “the unavailability/scarcity of resources (time, supplies, personnel, vehicles, transportation, and communication infrastructure) and the high stakes associated with delivering supplies (suffering and/or loss of life).” If we consider this, according to the official United Nations document, even before the earthquake, the transport infrastructure was very poor, it is easy to understand how challenging the delivery of relief items to beneficiaries can be.

One of the most important issues in the humanitarian field concerns the data. Usually, they are incomplete, scarce or completely missing. Typically, the severity of lack of data is higher in the post-disaster phase than in the development phase. For this reason, we place the research in the R-HL stage.

The case study is based on data present in documents received by the WFP association, and data present on the logistic cluster website and derived from one-to-one interviews with WFP operators in Italy and WFP logistics operators in Haiti. This research uses data on the demand for two kinds of item and the features of the considered vehicles (truck and helicopter) used in the distribution. Furthermore, it uses information about the aid distribution in Haiti.

The Republic of Haiti is a Caribbean country and it is divided into ten departments for reasons of administration. This case is focussed on the south-east department, whose capital is Jacmel, one of the areas most affected by the earthquake. Two independent kinds of LDC have been considered: the LCD for the trucks is in Jacmel, while the helicopter one, for emergency flight, is situated in Port au Prince. The municipalities – and the respective districts – considered as demand points, are Bainet (A), La Vallèe (B), Trouin (C), and Côtes-de-Fer (D). The situation and data are presented in Figure 1 and Table I.

Following the two-phase approach introduced by Balcik et al. (2008), in the first phase all available routes are defined for each kind of vehicle, considering the hypothesis of a four nodes distribution (Figure 1). All the routes are known a priori and are based on road network data.

Afterwards, the expected sets of vehicles, for each kind, have been estimated using formulas (1) and (2). For example, using formula (1) with demand of 90.937 kg for item 1 and 54.863 kg for item 2, average route for trucks equal to 334 minutes (it considers load and download time), M _k minutes in a day equal to 647 minutes (480 minutes plus half of the average route), and capacity of vehicles is 10,000 kg, the minimal theoretical number of vehicles, respectively, for item 1 is equal to ⌈4.69⌉=5, for item 2 is ⌈2.83⌉=3.

For helicopters, the average route is equal to 150 minutes and M _h minutes in a day equal to 555 minutes (480 minutes plus half of the average route), while in this case capacity of vehicles is 3,000 kg. Then, the minimal theoretical number of vehicles respectively for item 1 is equal to ⌈8.19⌉=9, for item 2 is ⌈4.94⌉=5.

In the case of co-transportation strategy, the minimal numbers of vehicle are respectively equal to ⌈7.52⌉=8 for trucks and ⌈13.13⌉=14 for helicopters.

This number gives an idea of the number of vehicles for the routing problem without the need for accurate detailed data. It could be possible to consider even the probability to use each route, seen in function of the needs per each node, in order to perform more effectively the results of the formula.

Finally, in the first phase, the costs associated with each kind of vehicle are estimated by several interviews as 1.50 /km for the trucks, and 10 /km for the helicopters.

Continuing the two-phase approach, starting from the list of candidate routes and the related travel times, the available number of vehicles, and the associated cost, in the second stage the modified models introduced in the previous section have been applied considering the items demands of each location, the volume capacity, and the available time for each vehicle under no-shortage constraints.

The application of the model under no-shortage constraint has underlined that, with an average speed of 50 km/h, the optimal solution to deliver the item 1 is performed by five trucks and 0 helicopters with 1.532,25 /day of total cost, while to deliver item 2 the optimal solution is 4 trucks and 0 helicopters with 1.270,5 /day of total cost (Figures 2 and 3). The model is solved using GAMS/Cplex on a PC equipped with an Intel CORE i3processor and Windows 7. The following section shows the graphics of the solutions in function of the available vehicles where the cost arises with the variation of number of available trucks and helicopters, considering always a 0 percent of shortage, because the two kinds of vehicles have different costs per km. In this application, it is possible to note that the most important data are regarding the distribution of the demand, calculated from official documents and the distribution of the population in the disaster area, the position of the depots, taken from official documents and maps, the types of vehicle delivered in this kind of situation, and all the available routes for the vehicles.

Figures 2 and 3 show two examples, for different items, of the increase in costs relative to the types of vehicle used. The increase is not always linear because in some situations the organization needs to use more helicopters to satisfy the lack of an available truck. The first column of each example considers that the delivery is done with trucks; it shows how many trucks should be used to cover the demand. The number of trucks decreases, while the number of helicopters increases. This means a non-linear growth per truck decrease.

According to the answers received from some logistics experts in Haiti, usually the distribution of different products is made separately by the organizations that are employed, as leaders of the clusters, for example, in the distribution of those kinds of products. Therefore, in this case, as previously explained, UNICEF supplies hygiene kits, while WFP delivers food. In this section we focus on the co-transportation of both food and hygiene kits. The results are presented in Figure 4.

The graphic shows that it is possible to use just seven trucks with 2,463.75 of total cost to deliver both the products, while if we consider two different transportations, it requires nine trucks with 2,802.75 of total cost (these data come from the previous example).

This result leads to a consideration of the co-transportation in order to have a higher level of cost performance even if, as suggested by Rodríguez et al. (2007), there are some other factors that can limit the co-transportation. According to Rodríguez et al. (2007), these factors can be the volume, weight, shape, and fragility of the items and should be checked. On the other hand, the benefits concern the minimization of “transportation costs and increase delivery service” (Hageback and Segerstedt, 2004). These results are obtained from the model in terms of reduction of costs and a higher performance in service level. Indeed, the total cost of co-transportation is lower than the sum of two different transportation ones and the total number of needed vehicles is lower. This means it is possible to have a higher level of service with the same size of fleet.

In this case, the application of the model under limited capacity constraints allows noting how the total cost and the shortage can change in function of the number of available trucks and helicopters. Figures 5 and 6 concern Food and Hygiene Kits, respectively.

These graphics give us an idea of the marginal costs associated with a service level reduction. Moreover, it suggests what the best option can be in terms of costs, considering the same level of shortage. According to some interviewed humanitarian operators, if just cost considerations are made, the best result is achieved using trucks, since the humanitarian organizations usually work under economic constraints. But in some cases, it could be useful to consider helicopters, when the distance or the accessibility to the location requires too much time or if it is not reachable by road. Furthermore, Figures 5 and 6 show the impact of the logistic system and of the fleet on the shortage.

The variables in these two figures (Figures 5 and 6) are the number of available vehicles. The number of vehicle varies from the maximum number of trucks to zero, defined using formula (1). Moreover, in each section of the pictures the number of available helicopters varies from zero to the maximum number of necessary helicopters needed to satisfy all the demands, using formula (2). These variations help the reader to understand how the shortage and the costs vary in function of the available fleet.

According to Kovács and Spens (2007), the research field of humanitarian logistics is relatively new, which, as Van Wassenhove and Pedraza Martinez (2012) underline, faces problems that have been successfully studied in the past. The purpose of the present work was to investigate the last mile distribution problem by providing an analytical model and a case study to assist relief decision makers in effective and efficient distribution across the last mile.

Initially, the first introduced analytical model is an extension of the model developed by Balcik et al. (2008). It was modified to consider the distribution problem relating on the variation of outputs, no-shortage constraint, and cost in function of the number and type of vehicles in the fleet.

Then, the basic approach composed of two phases introduced by the Balcik et al. (2008) model was used. In the first phase, the required data were collected and estimated, such as the demands, the route network, and the theoretical number of vehicles. In the second phase, the extended model was applied under no-shortage constraint.

Finally, relaxing this last restriction, the limited capacity of vehicles and its impact was studied using the second model above introduced.

The results obtained show that the fleet costs increase not linearly in function of the number of helicopters an organization needs to satisfy the lack of trucks. Furthermore, the research reveals that the variation in available fleet affects shortage and costs. These models can be applied in real operations in order to show how much can be saved in function of the considered vehicle typology and then the service level, defined as shortage. This leads to understand what the best way would be in processing the distribution with the lowest level of cost and which could be the implication in terms of cost and shortage in conditions of limited resources capacity.

Moreover, this research computed the performance of the delivery system when a co-distribution of different kinds of products is applied. These levels of performance are the same as those suggested by Rodríguez et al. (2007) and Hageback and Segerstedt (2004). The kinds of products considered in this case study are homogeneous in dimension and shape. Moreover, additional considerations, such as the presence of standard and modular packaging systems or the collaboration between humanitarian clusters, should be included and further discussed to determine the real feasibility of a co-transportation.

The main limitation associated to this research concerns the partial available data that can be collected during the operations. Moreover the complexity of the problem leads to some limitations in the mathematical models because solution times increase when the problems become larger and also when the resources are more limited.

In conclusion, it can be stated that although humanitarian logistics has its distinct features, the basic principles of business logistics can be successfully applied (Kovács and Spens, 2007).

As suggested by humanitarian operators during the interviews, an approach like the one in this research could be applied to different scenarios so to understand how they can differ and how different situations bring to different levels of performance of the distribution system. In this context, other applications can give an indication of what performance can be achieved with different levels of available logistics assets or in different scenarios.

Finally, another interesting application may consider how the errors in demand dislocation can impact costs and shortage. Indeed, as it has been underlined in the paper and in the literature, the data for humanitarian operations are often scarce or incomplete. For this reason, it is very interesting the impact of errors in demand estimations on costs or shortage.